[Math] Solving matrix AX=B matrix for X

Question

I'm having the following matrices in the form A X= B.Where A & B matrix are known values. How can i solve for the matrix X?. Please provide me some formulas or Ideas for solving this.

Note: we need to use the least square method to get the best solution for X $$\left[\begin{array}{ccc}A1&B1&C1\\A2&B2&C2\\…\\A10&B10&C10\end{array}\right]\left[\begin{array}{c}x1&x2&x3\\y1&y2&y3\\z1&z2&z3\end{array}\right]=\left[\begin{array}{c}A11&B11&C11\\A22&B22&C22\\…\\A1010&B1010&C1010\end{array}\right]$$

Can i use this formula for solving this?

$X= A^{-1}B$

Answer 1

There are a few options here:

1. Solve $A\pmatrix{x_1\\y_1\\z_1} = \pmatrix{A_{1,1}\\\vdots\\A_{10,10}}$ et cetera as three LS-problems each with vector unknows
2. Convert $AX=B$ to a vector unknown system $$\pmatrix{A&0&0\\0&A&0\\0&0&A} \pmatrix{x_1\\y_1\\z_1\\x_2\\y_2\\z_2\\x_3\\y_3\\z_3} = \pmatrix{A_{1,1}\\\vdots\\A_{10,10}\\B_{1,1}\\\vdots\\B_{10,10}\\C_{1,1}\\\vdots\\C_{10,10}}$$
3. Use the same formula for the solutions of (1.) to solve the three in parallel $$X = A^\dagger B = (A^TA)^{-1}A^T B$$